Rheology and Rheometry

Paula Moldenaers

Department of Chemical Engineering

Katholieke Universiteit Leuven

W. De Croylaan 46, B-3001 Leuven

Tel. 32(0)16 322675 Fax 32(0)16 322991

http://www.cit.kuleuven.ac.be/cit.

ρειν = flow

Rheology = The Science of Deformation and Flow

Why do we need it?- measure fluid properties- understand structure-flow property relations- modelling flow behaviour- simulate flow behaviour

of melts under processing conditions

Processing conditions shear rate [s-1]

Sedimentation 10-6 –10-4

Leveling 10-3

Extrusion 100-102

Chewing 101 -102

Mixing 101-103

Spraying, brushing 103 -104

Rubbing 104 -105

Injection molding 102 -105

coating flows 105 -106

How about Newtonian behaviour?

1. Constant viscosity

2. No time effects

3. No normal stresses in shear flow

4. η(ext)/η(shear) = 3

Symbols: ε° ⇒ Dσ ⇒Ts ⇒ σ

Newton’s law: T = -pI +η 2D

Contents

1. Rheological phenomena

2. Constitutive equations2.1. Generalized Newtonian fluids2.2. Linear visco-elasticity2.3. Non-linear viscoelasticity

3. Rheometry

4. Parameters affecting rheology

1. Rheological Phenomena:

Do real melts behave according to Newton’s law?

Deviation 1:

Mayonnaise: resistance (viscosity) decreases with increasing shear rate: shear thinning

Starch solution: resistance increases with increasing shear rate: shear thickening

This is a non-linearity:

)(γη &

Do real melts behave according to Newton’s law?

G(t) or G(t,γ)

Deviation 2: example silly putty

The response of the material depends on the time scale:* Short times: elastic like behaviour * Long times: liquid-like behaviour

VISCO-ELASTIC BEHAVIOUR

t > τt < τ

Linear non-linear

Weissenberg effect Die Swell

Normal stresses

Do real melts behave according to Newton’s law?Deviation 3:

Do real melts behave according to Newton’s law?

deviation 4: Ductless Syphon

η(ext) > > η(shear)

In summary: Newtonian behaviour real behaviour

1. Constant viscosity 1. Variable viscosity

2. No time effects 2. Time effects

3. No normal stresses in shear flow 3. Normal stresses

4. η(ext)/η(shear) = 3 4. Large η(ext)

3 dim: T = -pI + η (2D) ?Simple shear: σ = η dγ/dt

Contents

1. Rheological phenomena

2. Constitutive equations2.1. Generalized Newtonian fluids2.2. Linear visco-elasticity2.3. Non-linear viscoelasticity

3. Rheometry

4. Parameters affecting rheology

2. Constitutive equations2.1 Generalized Newtonian fluids

Examples of non-Newtonian behaviour:

Macosko, 1992

2.1 Generalized Newtonian fluids

Non-Newtonian behaviour is typical for polymeric solutions and molten polymers

e.g. ABS polymer melt (Cox and Macosko)

2.1 Generalized Newtonian fluids

DI D 2)II(fpT 21 ⋅+−=

The viscosity is now replaced by a function of the second invariant of 2D

for a shear flow this becomes:

and different forms for this function have been proposed

( )σ η γ γxy = ⋅12& &

With: II2D = 1/ 2 (tr22D – tr (2D)2)

tr = trace = sum of the diagonal elements

Model 1: Power Law

1nnxy

2/)1n(2

kk

2IIkpT

−

−

γ=ηγ=σ

⋅⋅+−==

&&

DI D

2 parameters

shear rate (s-1)

0.001 0.01 0.1 1 10 100 1000η

0.001

0.01

0.1

1

10

100

n=0.4, k=1n=0.8, k=1n=1.4, k=0.01

shear rate (s−1)

0 200 400 600 800 1000 1200

σ

0

50

100

150

200

250

300

n<1 : shear-thinningn>1: shear-thickening

Model 2: Ellis model

( )

ηη γ

ηη

σ

0

0

11

1

1

1

=+ ⋅

= +⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

−

−

k

II

k

n

n

& ( ) 3 parameters

shear rate (s-1)

0.001 0.01 0.1 1 10 100 1000

η

0.1

1

10

100

1000

log η0

n-1

Model 3: CROSS model

η ηη η γ

−−

=+ ⋅

∞

∞ −0

11

1 k n& ( ) 4 parameters

shear rate (s-1)

1e-3 1e-2 1e-1 1e+0 1e+1 1e+2 1e+3 1e+4 1e+5

η

0.1

1

10

100

1000

log η0

n-1

log η∞

( )η η

η η−−

=+ ⋅

∞

∞−

0

1

1 22

1 2k II

nD

( )/

3D

Special case: plastic Behaviour

σ σ γ σ γ

σ σ γ

< → = = ⋅

> → ≠

y

y

G& ,

&

0

0

No flow, only deformation Yield stress

What have we gained in generalized Newtonian?

Newtonian behaviour real behaviour

1. Constant viscosity 1. Variable viscosity

2. No time effects 2. Time effects

3. No normal stresses in shear flow 3. Normal stresses

4. η(ext)/η(shear) = 3 4. Large η(ext)

3 dim: T = -pI + η (2D) T = -pI + η(II2D) (2D)

Simple shear: σ = η dγ/dt

Contents

1. Rheological phenomena

2. Constitutive equations2.1. Generalized Newtonian fluids2.2. Linear visco-elasticity2.3. Non-linear viscoelasticity

3. Rheometry

4. Parameters affecting rheology

2. Constitutive equations2.2 Linear visco-elasticity

Reference: 2 extremes

Hooke’s Law Newton’s Law(solid mechanics) (fluid mechanics)

σ = G.γ σ = η .dγ/dt

G = modulus (Pa) η = viscosity (Pa.s)Material property material property

Time effects (linear visco-elastic phenomena):

Example 1: creep

Apply constant stress σ

Compliance

J t t( ) ( )=

γσ0

Time effects (linear visco-elastic phenomena):

Example 2: stress relaxation upon step strain

Apply constant strain

Modulus

G(t) = σ(t)/γ

Example for molten LDPE [Laun, Rheol. Acta, 17,1 (1978)]

How to descibe this behaviour?Example: differential models

Phenomenological models

Hookean spring

Newtonian dashpot

101 G γ=σ

202 γ⋅η=σ &Maxwell Kelvin-Voigt

γ γ γ

σ σ σ

γ γ γ

γσ σ

η

ση

σ η γ

σ τσ η γ

= +

= =

= +

= +

+⎛⎝⎜

⎞⎠⎟ =

+ =

1 2

1 2

1 2

1

0

2

0

0

00

0

& & &

&&

& &

& &

G

G

Maxwell model:

τ = relaxation time

ω [rad/s]

0.01 0.1 1 10

G(t)

0

2

4

6

8

10

12

τ

G0

Example: Stress relaxation upon step strain for a Maxwell element

γ γ

σ τσ

σ γ

σγ

τ

= ≥

+ =

= =

= = ⋅ −

0

0 0

0

0

0

0

t

ddt

t G

t G t G t

;

( ) ( ) exp( / )

Generalized Maxwell model to describe molten polymers:

σ σ

σ τ σ η γ

τ

TOT ii

i i i i

ii

iG t G t

= ∑

+ =

= −∑

& &

( ) exp( / )0

Relaxation functions:For a simple Maxwell model: single exponential (single relaxation time τ):

G t G t

G t G t

G t F t d

G t H t d

i i

( ) exp( / )

( ) exp( / )

( ) ( )exp

( ) ( )exp

= −

= −∑

=−⎛

⎝⎜⎞⎠⎟∫

=−⎛

⎝⎜⎞⎠⎟∫

∞

∞

0

0

0

τ

τ

ττ

τ

ττ

ττ

For a generalized Maxwell fluid (discrete number of relaxation times τI:

We can replace the discrete relaxation times by a continuous spectrum:

Or based on a logaritmic time scale : the relaxation spectrum is defined by:

Time effects (linear visco-elastic phenomena): Example 3: Oscillatory experiments

STRAIN

t

i t

STRAIN RATE

t t

i t

HOOKEAN SOLID

G G tG i t

NEWTONIAN FLUID

ti i t i

:

sin( )

exp( )

& cos( ) sin( )

& i exp( )

sin( )exp( )

& sin( )exp( )

γ γ ω

γ γ ω

γ γ ω ω γ ω

γ ωγ ω

σ γ γ ωσ γ ω

σ ηγ ηγ ω ωσ η γ ω ω η γω

=

=

= = + °

=

= ==

= = + °= =

0

0

0 0

0

00

00

90

90

[ ]( ) ( )[ ]

VISCOELASTIC MATERIAL

G i

G i

complex ulus

G

G t

G t tG t G t

G t G t

G iG

el viscσ σ σ

σ γ ωηγ

σ ωη γ

σ γ

σ γ ω δ

σ γ ω δ ω δσ δ γ ω δ γ ω

σ ω ω γ

σ γ

= +

= +

= +

=

= +

= += ⋅ + ⋅

= ⋅ + ⋅

= +

∗

∗

∗

∗ ∗

( )

mod :

sin( )

sin( ).cos( ) cos( ).sin( )cos( ) sin( ) sin( ) cos( )

' sin( ) " cos( )

( ' " )

0

0

0 0

0

Storage and Loss modulustan ' ''

δ =GG

Dynamic moduli of a Maxwell fluid

w [rad/s]

0.01 0.1 1 10 100

Mod

uli [

Pa]

0.001

0.01

0.1

1

10

G'G"

G0

τG G

G G

'( )

"( )

ωω τω τ

ωωτ

ω τ

=+

=+

02 2

2 2

02 2

1

1

G’ and G” for a generalized Maxwell fluid:

G G s s ds G e s ds

G

G G s s ds

G

iis

i

N

ii

ii

N

ii

ii

N

" ( ) cos( ) cos( )

( )

' ( ) sin( )

( )( )

= ⋅ =⎛⎝⎜

⎞⎠⎟

⋅

=+

= ⋅

=+

∞−

=

∞

=

∞

=

∫ ∑∫

∑

∫

∑

ω ω ω τ ω

ωτωτ

ω ω

ωτωτ

0 10

21

0

2

21

1

1

What have we gained in linear visco-elasticity? Newtonian behaviour real behaviour

1. Constant viscosity 1. Variable viscosity

2. No time effects 2. Time effects

3. No normal stresses in shear flow 3. Normal stresses

4. η(ext)/η(shear) = 3 4. Large η(ext)

3 dim: T = -pI + η (2D) G(t), H(τ),…Simple shear: σ = η dγ/dt fully descibes linear VE

Contents

1. Rheological phenomena

2. Constitutive equations2.1. Generalized Newtonian fluids2.2. Linear visco-elasticity2.3. Non-linear viscoelasticity

3. Rheometry

4. Parameters affecting rheology

2. Constitutive equations2.3. Non-linear visco-elasticityExample 1: Steady state shear flow

22

21

γσσ

γσσ

γσ

η

&

&

&

zzyy

yyxx

xy

−=Ψ

−=Ψ

=

zzyy

yyxx

xy

N

N

σσ

σσ

σ

−=

−=

2

1

LDPE, Laun et al.PIB in decalin, Keentok et al.

Example 2: Non-linear stress relaxation upon step strain

LDPE, Laun et al.

Example 3: Stress evolution upon inception of shear or elongational flow

What have we gained in non-linear visco-elasticity? Newtonian behaviour real behaviour

1. Constant viscosity 1. Variable viscosity

2. No time effects 2. Time effects

3. No normal stresses in shear flow 3. Normal stresses

4. η(ext)/η(shear) = 3 4. Large η(ext)

3 dim: T = -pI + η (2D) no universal model

Simple shear: σ = η dγ/dt

Contents

1. Rheological phenomena

2. Constitutive equations2.1. Generalized Newtonian fluids2.2. Linear visco-elasticity2.3. Non-linear viscoelasticity

3. Rheometry

4. Parameters affecting rheology

3. Rheometry

Why ? 1. Input for Constitutive Equations2. Quality control3. Simulate industrial flows

A rheometer is an instrument that measures both stress and deformation.

indexerviscometer

What do we want to measure?- steady state data- small strain (LVE) functions- large strain deformations

Introduction: classifications

Kinematics : shear vs elongation

homogeneous vs non-homogeneous vs complex flow fields

type of straining:

- small : G’(ω), G”(ω), η+(t), η-(t), G(t), σy- large : G’(ω,γ), G”(ω,γ), η+(t,γ), η-(t ,γ), G(t ,γ), η(t,ε)- steady : η( ), Ψ1( )

shear rheometry : Drag or pressure driven flows.

&γ &γ

Shear flow geometries

Pressure driven flowsDrag flows

Sliding plates

Couette

Cone and Plate

Parallel plates

Capillary

Slit

Annulus

Macosko, 1992

Drag flows : Cone and plate

( )

( )

rr

r

r rvr

r r r

r r

rr

r

:

:sin

sin cot

: cot

1

1 0

1 2 0

2

2 2∂ σ

∂

σ σρ

θθ

∂ σ θ

∂θ

σ

φ∂σ

∂θθ σ

θθ φφ θ

θθθ

θφθφ

−+

= −

− ⋅ =

+ ⋅ =

Probably most popular geometry Mooney (1934)

1. Steady, laminar, isothermal flow2. Negligible gravity and end effects3. Spherical boundary liquid4. vr=vz=0 and vφ(r,θ)5. Angle α < 0.1 radians

Equations of motion:

Drag flows : Cone and plate geometry

Boundary conditions1. vφ(π/2) = 02. vφ(π/2-α)=ωrsin(π/2-α)≈ωr

Shear stress

( )φ

σ

θφ σ σ

θ

σ φ

σπ

θφθφ θφ

θφ

π

θϕ

: cotsin

1 2 03

2

2

2

00

23

r

d

d rC Cte

M r drd

Mr

R

+ ⋅ = → = ≅

= ∫∫

⎫

⎬

⎪⎪⎪

⎭

⎪⎪⎪

=⋅

Independent of fluid characteristics because of small angle!

Drag flows : Cone and plate geometry

Shear rate: is also constant throughout the sample: homogeneous!

&( ) ( ) ( )

γω

αωα

ωα

= =⋅

⋅= ≅

vh r

rr tg tg

r

Normal stress differences: total trust on the plate is measured Fz

( )F R

N

z

FR

z

= −

=

πσ σθθ φφ

π

2

12

2

2

Drag flows : Cone and plate geometry

+ - constant shear rate, constant shear stress -homogeneous!- most useful properties can be measured- both for high and low viscosity fluids- small sample- easy to fill and clean

- - high visc: shear fracture - limits max. shear rate-(low visc : centrifugal effects/inertia - limits max. shear rate)- (settling can be a problem)- (solvent evaporation)- stiff transducer for normal stress measurements- viscous heating

Drag flows : Parallel plates

Again proposed by Mooney (1934)

1. Steady, laminar, isothermal flow2. Negligible gravity and end effects3. cylindrical edge4. vr=vz=0 and vφ(r,z)

Equations of motion:

( )

( )

( )

rr

rr r

vr

z

zz

rr

z

zz

:

:

:

1

0

0

2∂ σ

∂σ

ρ

θ∂ σ

∂

∂ σ

∂

θθ θ

θ

− = −

=

=

Drag flows : Parallel plates

Shear rate: is not constant throughout the sample

&γω

= =⋅v

hr

hr

Shear stressσ

π γ= +

⎡

⎣⎢

⎤

⎦⎥

MR

d Md R2

13lnln &

Normal stressesN N

FR

d Fzd

z

R1 2 2 2− = +

⎡

⎣⎢

⎤

⎦⎥

π γlnln &

Drag flows : Parallel plate geometry

+ - preferred geometry for viscous melts - small strain functions- sample preparation is much simpler- shear rate and strain can be changed also by changing h- determination of wall slip easy- N2 when N1 is known- edge fracture can be delayed

- - non-homogeneous flow field (correctable)- inertia/secondary flow - limits max. shear rate- edge fracture still limits use- (settling can be a problem)- (solvent evaporation)- viscous heating

Pressure driven flows : capillary rheometry

1. Steady, laminar, isothermal flow2. No slip at the wall, vx =0 at R=03. vr=vθ=04. Fluid is incompressible, η≠f(p)

Equation of motion:r

Z

θ

L

RQ

zr

rr

pz

rz: ( )1 0∂ σ∂

∂∂

− =

Only a function of rOnly a function of z

1r

d rdr

dpdz

rz( )σ=

1

2

0 0

2

2

2

rd r

drP P

L

P PL

r Cr

C because r

RP P

LR

r rR

rz o L

rzo L

rz

rz wo L

rz w

( )

@

( )

( )

σ

σ

σ

σ σ

σ σ

=−

=−

⋅ +

= ≠ ∞ =

= =−

⋅

=Note : independent of fluid properties

( )

Q v r rdr

Q v r r

r R and dr R d

Q r R R d

QR

d

zR

zR dv

drdr

R

w w

dvdr

drR

w

dvdr

w

w dvdr

z

z zw

zw

= ∫

= − ∫

= =

= − ∫ = −⎛⎝⎜

⎞⎠⎟∫

= − ∫

2

2 2

2 2

0

02

0

2

0

2

0

3

32

0

π

π π

σσ

σσ

π πσ

σσ

σ

σ

πσ σ

σ

σ

( )

3

34

34

43

2

3

3

32

3 3

QR R

dQd

QR

QR

d Qd

d Qd

w w

ww

dvdr

wdvdr w

wa

w

z

w

z

w

σ

π

σ

π σσ

γπ π σ

γγ

σ

σ

σ

+ ⋅ = −

= + ⋅

= +⎛⎝⎜

⎞⎠⎟

− =&ln

ln

&& ln

ln

Integration by parts + assume :no slip

substitute r and dr

Typical “trick” to deal with inhomogeneous flows:changing of variables

Shear rate calculation from Q

Differentiate with respect to σwusing Leibnitz’s rule

r/R

0 1

v z / <

v z>

0

1

2

n=1n=0.7n=0.3n=0.01

γγ

σwa

w

d Qd

= +⎛⎝⎜

⎞⎠⎟

& lnln4

3Weissenberg-Rabinowitsch “Correction”

“Correction” factor accounts for material behaviour

Physical meaning:

e.g. power law fluid

Steepness of the velocity profile changesShear rate is increased with respect to the Newtonian case:

γγ

wa

n= +⎛

⎝⎜⎞⎠⎟

&

43 1

&γπ

aQR

=4

3

What pressure drop do we really measure?

2R

LPP Lo

w ⋅−

=σCannot be measured!

z=0 z=LCapillairyReservoir

Exit

∆Pen

∆P ∆Ptot

∆Pex

L/D

0 10 20 30 40 50 60

∆p

[MPa

]

0

10

20

30

40

50

10s-1 30s-1 50s-1 100s-1 200s-1 300s-1 500s-1

BAGLEY plots

L/De

Pressuredrop

γ3

γ1

γ2

∆Pe

LDPE @190°C

)eRL(2RP

w +⋅∆

=σ ∆pe can be used to estimatethe elongational viscosity (contraction flow)

Are these conditions met ? 1. Steady, laminar, isothermal flow2. No slip at the wall, vx =0 at R=03. vr=vθ=04. Fluid is incompressible, η≠f(p)

Problem 1: Melt distortion

Melt fracture typically occurs at σw 105 Pa

Are these conditions met ? 1. Steady, laminar, isothermal flow2. No slip at the wall, vx =0 at R=03. vr=vθ=04. Fluid is incompressible, η≠f(p)

Problem 2: Viscous heating

NaRk= ατ γ2

4

.

Are these conditions met ? 1. Steady, laminar, isothermal flow2. No slip at the wall, vx =0 at R=03. vr=vθ=04. Fluid is incompressible, η≠f(p)

Problem 3: Wall slip

cst@Rv4 s

a =σ+γ=γ &&

Are these conditions met ? 1. Steady, laminar, isothermal flow2. No slip at the wall, vx =0 at R=03. vr=vθ=04. Fluid is incompressible, η≠f(p)

Problem 4: Melt compressibility, η=f(p)

L/D

0 10 20 30 40 50 60

Pres

sure

dro

p(M

Pa)

0

10

20

30

40

50

60

70

80

10 s-1

30 s-1

50 s-1

100 s-1

200 s-1

300 s-1

500 s-1

1000 s-1

2000 s-1

Capillary rheometry : conclusions

+ - Simple yet accurate!- High shear rates possible- Sealed system, can be pressurized- Process simulator- Entrance flows and exit flows can be used- MFI

- - Non-homogeneous flow field (correctable)- Only viscosity data, some indications for N1, ηe- Lot of data required (Bagley plots)- Melt fracture limits shear rate- Wall slip can be a problem- Viscous heating- Shear history / degradation

Melt Flow index

Shear rheometers: the verdict

Couette + low η, high rates - end corrections+ can be homogeneous - high visc. : too difficult+ settling - no N1

Cone and Plate + best for N1 - edges, low shear rate+ homogeneous - free surface+ transient meas. - alignment

Parallel Plate + easy to load - edges+ G’,G” for melts - non-homogenous+ vary h! - free surface

Capillary + high rates - corrections:+ accuracte - non-homogenous+ sealed - no N1

Contents

1. Rheological phenomena

2. Constitutive equations2.1. Generalized Newtonian fluids2.2. Linear visco-elasticity2.3. Non-linear viscoelasticity

3. Rheometry

4. Parameters affecting rheology

4. Parameters affecting rheology

• Chemistry• Molecular weight• Molecular weight distribution• Molecular architecture (branching) • Fillers/additives• Temperature• Pressure•…

EEffect of molecular weight on the viscosity curve

Example: narrow MW polystyrenes

From Stratton

4.3wM∝η

PS, 217°C

2.Mw,e=39000

wM∝η

Effect of the MW on the zero shear viscosity for linear molten polymers

Log η+ ct

Log Mw + ctBerry and Fox

Effect of molecular weight on moduli⎢

8900Mw =

581000Mw =

Onogi et al., 1970

Effects of molecular weight distribution

Viscosity:

Shear thinning sets in earlier with increasing molecular weight distribution

From Uy and Greassley

EEffect of chain architecture (branching)Low shear rates higher shear rates

O = linear

and ∆ = branched

Kraus and Gruver

EEffect of fillers

Chapman and Lee

EEffect of temperature: time-temperature superposition

-C1 (T-Tr)

WLF-relation: log aT = ------------

C2 + T - Tr

Effect of pressure on viscosity

Relevant for injection moulding

Add-on ‘pressure chamber’ on Gottfert capillary rheometer

TdPd

⎟⎠⎞

⎜⎝⎛=

ηβ ln

“Corrected” viscosity for PMMA (210° C)

Shear rate (s-1)

10 100

Visc

osity

(Pas

)

1e+3

1e+4

80 MPa70 MPa55 MPa40 MPa30 MPa0.1 MPa

Mean pressures:

Viscoity curves for PαMSAN and LDPE

Shear rate (s-1)

10 100

Visc

osity

(Pas

)

1e+3

1e+4

80 MPa70 MPa55 MPa40 MPa30 MPa0.1 MPa

Mean pressures:

PαMSAN

LDPE

Determination of β at several shear rates

Pmean (MPa)0 20 40 60 80 100 120

ln η

7

8

9

10

11

6 s-1

13 s-1

33 s-1

70 s-1

108 s-1

146 s-1

223 s-1

260 s-1

290 s-1

Shear rates:

TdPd

⎟⎠⎞

⎜⎝⎛=

ηβ ln

β at several shear rates

Shear rate (s-1)

10 100 1000

β γ(GPa

-1)

2

4

6

8

10

12

14

16

18PMMAPαΜSANLDPEPMMAPαMSANLDPE

Determination of β at several shear stresses (PMMA)

Pmean (MPa)0 20 40 60 80 100 120

ln η

6

7

8

9

10

11

80 kPa100 kPa140 kPa206 kPa270 kPa363 kPa430 kPa500 kPa

Stresses:

TdPd

⎟⎠⎞

⎜⎝⎛=

ηβ ln

β at several shear stresses (PMMA)

Shear stress (kPa)0 100 200 300 400 500 600

βσ(

GP

a-1)

5

10

15

20

25

30

35

40

45

PMMAPαMSANLDPE

Useful references concerning rheology

Rheology: Principles, Measurements and Applications

C. Macosko Ed, VCH (1994)

Understanding Rheology

F. Morrison, Oxford University Press (2001)

Structure and Rheology of Molten Polymers

J.M. Dealy and R.G. Larson, Hanser (2006)